A029958 Numbers that are palindromic in base 13.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 170, 183, 196, 209, 222, 235, 248, 261, 274, 287, 300, 313, 326, 340, 353, 366, 379, 392, 405, 418, 431, 444, 457, 470, 483, 496, 510, 523, 536, 549, 562

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• base
• easy

Comments

Cilleruelo, Luca, & Baxter prove that this sequence is an additive basis of order (exactly) 3. - Charles R Greathouse IV, May 04 2020

Links

• John Cerkan, Table of n, a(n) for n = 1..10000
• Javier Cilleruelo, Florian Luca and Lewis Baxter, Every positive integer is a sum of three palindromes, Mathematics of Computation, Vol. 87, No. 314 (2018), pp. 3023-3055, arXiv preprint, arXiv:1602.06208 [math.NT], 2017.
• Patrick De Geest, Palindromic numbers beyond base 10.
• Phakhinkon Phunphayap and Prapanpong Pongsriiam, Estimates for the Reciprocal Sum of b-adic Palindromes, 2019.
• Index entries for sequences that are an additive basis, order 3.

Crossrefs

Palindromes in bases 2 through 12: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957.

Programs

Mathematica

f[n_,b_]:=Module[{i=IntegerDigits[n,b]},i==Reverse[i]];lst={};Do[If[f[n,13],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 08 2009 *)
Select[Range[0,600],IntegerDigits[#,13]==Reverse[IntegerDigits[#,13]]&] (* Harvey P. Dale, Nov 16 2022 *)

PARI

isok(n) = my(d=digits(n, 13)); d == Vecrev(d); \\ Michel Marcus, May 13 2017

Python

from sympy import integer_log
from gmpy2 import digits
def A029958(n):
    if n == 1: return 0
    y = 13*(x:=13**integer_log(n>>1,13)[0])
    return int((c:=n-x)*x+int(digits(c,13)[-2::-1]or'0',13) if nChai Wah Wu, Jun 14 2024

Formula

Sum_{n>=2} 1/a(n) = 3.55686013... (Phunphayap and Pongsriiam, 2019). - Amiram Eldar, Oct 17 2020